{"id":1397,"date":"2024-09-17T17:24:42","date_gmt":"2024-09-17T17:24:42","guid":{"rendered":"https:\/\/gbelentzas.sites.sch.gr\/?p=1397"},"modified":"2024-09-20T04:14:51","modified_gmt":"2024-09-20T04:14:51","slug":"1-1-%ce%b3-%cf%84%ce%b5%cf%84%cf%81%ce%b1%ce%b3%cf%89%ce%bd%ce%b9%ce%ba%ce%ae-%cf%81%ce%af%ce%b6%ce%b1-%cf%80%cf%81%ce%b1%ce%b3%ce%bc%ce%b1%cf%84%ce%b9%ce%ba%ce%bf%cf%8d-%ce%b1%cf%81%ce%b9%ce%b8","status":"publish","type":"post","link":"https:\/\/gbelentzas.sites.sch.gr\/?p=1397","title":{"rendered":"1.1.\u0393. \u03a4\u03b5\u03c4\u03c1\u03b1\u03b3\u03c9\u03bd\u03b9\u03ba\u03ae \u03c1\u03af\u03b6\u03b1 \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03bf\u03cd \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03cd"},"content":{"rendered":"
\n

<\/i> \u03a3\u03b7\u03bc\u03b5\u03b9\u03ce\u03c3\u03b5\u03b9\u03c2 \u0398\u03b5\u03c9\u03c1\u03af\u03b1\u03c2<\/em><\/span><\/h4>\n<\/div>\n

\u03a0\u03c9\u03c2 \u03bf\u03c1\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03b7 \u03c4\u03b5\u03c4\u03c1\u03b1\u03b3\u03c9\u03bd\u03b9\u03ba\u03ae \u03c1\u03af\u03b6\u03b1 \u03b5\u03bd\u03cc\u03c2 \u03b8\u03b5\u03c4\u03b9\u03ba\u03bf\u03cd \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03cd x;;<\/strong><\/span><\/em><\/p>\n

\u0397 \u03c4\u03b5\u03c4\u03c1\u03b1\u03b3\u03c9\u03bd\u03b9\u03ba\u03ae \u03c1\u03af\u03b6\u03b1 \u03b5\u03bd\u03cc\u03c2 \u03bc\u03b7 \u03b1\u03c1\u03bd\u03b7\u03c4\u03b9\u03ba\u03bf\u03cd \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03cd x \u03c3\u03c5\u03bc\u03b2\u03bf\u03bb\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03bc\u03b5 \"\sqrt{x}\" \u03ba\u03b1\u03b9 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bf \u03bc\u03b7 \u03b1\u03c1\u03bd\u03b7\u03c4\u03b9\u03ba\u03cc\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2 \u03b1 \u03c0\u03bf\u03c5 \u03cc\u03c4\u03b1\u03bd \u03c5\u03c8\u03c9\u03b8\u03b5\u03af \u03c3\u03c4\u03bf \u03c4\u03b5\u03c4\u03c1\u03ac\u03b3\u03c9\u03bd\u03bf \u03bc\u03b1\u03c2 \u03b4\u03af\u03bd\u03b5\u03b9 \u03c4\u03bf\u03bd \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc x.<\/p>\n

\"\"<\/p>\n

\u0393\u03b9\u03b1 \u03c0\u03b1\u03c1\u03ac\u03b4\u03b5\u03b9\u03b3\u03bc\u03b1, \"\sqrt{25} =5, \quad\" \u03b1\u03c6\u03bf\u03cd \"^2=25.\"<\/p>\n

\u0395\u03c0\u03af\u03c3\u03b7\u03c2, \u03bf\u03c1\u03af\u03b6\u03bf\u03c5\u03bc\u03b5 \u03cc\u03c4\u03b9 \"\sqrt{0} =0.\"<\/p>\n

<\/i> \u00a0\u03a0\u03b1\u03c1\u03b1\u03c4\u03ae\u03c1\u03b7\u03c3\u03b7:<\/strong> <\/em><\/span>\u0394\u03b5\u03bd \u03bf\u03c1\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c4\u03b5\u03c4\u03c1\u03b1\u03b3\u03c9\u03bd\u03b9\u03ba\u03ae \u03c1\u03af\u03b6\u03b1 \u03b1\u03c1\u03bd\u03b7\u03c4\u03b9\u03ba\u03bf\u03cd \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03cd, \u03b3\u03b9\u03b1\u03c4\u03af \u03b4\u03b5\u03bd \u03c5\u03c0\u03ac\u03c1\u03c7\u03b5\u03b9 \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2 \u03c0\u03bf\u03c5 \u03c4\u03bf \u03c4\u03b5\u03c4\u03c1\u03ac\u03b3\u03c9\u03bd\u03bf \u03c4\u03bf\u03c5 \u03bd\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b1\u03c1\u03bd\u03b7\u03c4\u03b9\u03ba\u03cc\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2.<\/p>\n

 <\/p>\n

\u0391\u03c0\u03bb\u03bf\u03c0\u03bf\u03af\u03b7\u03c3\u03b7 \u03c4\u03b5\u03c4\u03c1\u03b1\u03b3\u03c9\u03bd\u03b9\u03ba\u03ae\u03c2 \u03c1\u03af\u03b6\u03b1\u03c2 \u03bc\u03b5 \u03c4\u03bf “\u03c4\u03b5\u03c4\u03c1\u03ac\u03b3\u03c9\u03bd\u03bf”.<\/span><\/strong><\/p>\n

    \n
  • \u0391\u03bd \u03b1<\/i> \u03b5\u03af\u03bd\u03b1\u03b9 \u03bc\u03b7 \u03b1\u03c1\u03bd\u03b7\u03c4\u03b9\u03ba\u03cc\u03c2<\/b> \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2, \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae \"a \geq 0\", \u03c4\u03cc\u03c4\u03b5:\u00a0\"\left(\sqrt{\alpha}\right)^2 = \alpha\"<\/li>\n<\/ul>\n

    (\u0393\u03b9\u03b1 \u03b5\u03c5\u03ba\u03bf\u03bb\u03af\u03b1, \u03bc\u03c0\u03bf\u03c1\u03bf\u03cd\u03bc\u03b5 \u03bd\u03b1 \u03b3\u03c1\u03ac\u03c8\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b7\u03bd \u03c1\u03af\u03b6\u03b1 \u03c7\u03c9\u03c1\u03af\u03c2 \u03c4\u03b7\u03bd \u03c0\u03b1\u03c1\u03ad\u03bd\u03b8\u03b5\u03c3\u03b7, \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae \"{\sqrt{\alpha}}^2 = \alpha\").<\/p>\n

      \n
    • \u0391\u03bd \u03b1<\/i> \u03b5\u03af\u03bd\u03b1\u03b9 \u03bf\u03c0\u03bf\u03b9\u03bf\u03c3\u03b4\u03ae\u03c0\u03bf\u03c4\u03b5<\/b> \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2, \u03c4\u03cc\u03c4\u03b5: \"\sqrt{\alpha^2} = |\alpha|\"<\/li>\n<\/ul>\n

      \u0393\u03b9\u03b1 \u03c0\u03b1\u03c1\u03ac\u03b4\u03b5\u03b9\u03b3\u03bc\u03b1:<\/p>\n

        \n
      • \"{\sqrt{3}}^2 = 3\"<\/li>\n
      • \"\sqrt{3^2} = |3| = 3\"<\/li>\n
      • \"\sqrt{(-3)^2} = |-3| = 3\"<\/li>\n<\/ul>\n

         <\/p>\n

        \u03a0\u03bf\u03b9\u03b5\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bf\u03b9 \u03b9\u03b4\u03b9\u03cc\u03c4\u03b7\u03c4\u03b5\u03c2 \u03c4\u03c9\u03bd \u00a0\u03c4\u03b5\u03c4\u03c1\u03b1\u03b3\u03c9\u03bd\u03b9\u03ba\u03ce\u03bd \u03c1\u03b9\u03b6\u03ce\u03bd \u03b4\u03cd\u03bf \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03ce\u03bd \u03b1\u03c1\u03b9\u03b8\u03bc\u03ce\u03bd;<\/span><\/strong><\/em><\/p>\n

        \u0393\u03b9\u03b1 \u03b4\u03cd\u03bf \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03bf\u03cd\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03cd\u03c2 \u03b1, \u03b2 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 \u03cc\u03c4\u03b9:<\/p>\n

          \n
        • \"\sqrt{\gra\cdot\grb}=\sqrt{\gra}\cdot\sqrt{\grb}\" \u03bc\u03b5 \"\gra, \beta\geq 0.\"<\/li>\n
        • \"\sqrt{\dfrac{\gra}{\grb}}=\dfrac{\sqrt{\gra}}{\sqrt{\grb}}\" \u03bc\u03b5 \"\gra\geq 0, \grb>0.\"<\/li>\n<\/ul>\n

           <\/p>\n

          <\/i> \u00a0\u03a0\u03b1\u03c1\u03b1\u03c4\u03ae\u03c1\u03b7\u03c3\u03b7:<\/span> \u00a0<\/span><\/i><\/b>\u0391\u03bd \u03b1, \u03b2 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b8\u03b5\u03c4\u03b9\u03ba\u03bf\u03af<\/strong> \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03af, \u03b3\u03b9\u03b1 \u03c4\u03bf \u03ac\u03b8\u03c1\u03bf\u03b9\u03c3\u03bc\u03b1 \u03c4\u03b5\u03c4\u03c1\u03b1\u03b3\u03c9\u03bd\u03b9\u03ba\u03ce\u03bd \u03c1\u03b9\u03b6\u03ce\u03bd \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 \u03cc\u03c4\u03b9<\/p>\n

            <\/span>   <\/span>\"\[\sqrt{\alpha}+ \sqrt{\beta}\neq \sqrt{\alpha+\beta}.\]\"<\/p>\n

          \u0397 \u03c0\u03b1\u03c1\u03b1\u03c0\u03ac\u03bd\u03c9 \u03b9\u03c3\u03cc\u03c4\u03b7\u03c4\u03b1 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 \u03b1\u03bd\u00a0\u03b1=0 \u03ae \u03b2=0.<\/strong><\/p>\n

          \n

          <\/i> <\/i>Let’s Practise<\/b><\/span><\/h4>\n

          1. \u039d\u03b1 \u03b1\u03c0\u03bb\u03bf\u03c0\u03bf\u03b9\u03ae\u03c3\u03b5\u03c4\u03b5 \u03c4\u03b9\u03c2 \u03c0\u03b1\u03c1\u03b1\u03c3\u03c4\u03ac\u03c3\u03b5\u03b9\u03c2:<\/strong><\/em><\/span><\/p>\n

          \n
          \n

          \u03b1.<\/strong> \"\sqrt{72} + \sqrt{18}\"<\/p>\n

          \u03b2.<\/strong> \"\sqrt{\dfrac{45}{5}} + \sqrt{20}\"<\/p>\n

          \u03b3.<\/strong> \"3\sqrt{12} - \sqrt{27} + 2\sqrt{75}\"<\/p>\n

          \u03b4.<\/strong> \"\sqrt{32} - 2\sqrt{8} + \sqrt{50}\"<\/p>\n

          \u03b5.<\/strong> \"\sqrt{100} + \sqrt{45} - \sqrt{5}\"<\/p>\n<\/div>\n

          \n

          \u03c3\u03c4.<\/strong> \"\sqrt{\dfrac{81}{9}} + \sqrt{64}\"<\/p>\n

          \u03b6.<\/strong> \"2\sqrt{48} - 4\sqrt{12}\"<\/p>\n

          \u03b7.<\/strong> \"\sqrt{18} + 3\sqrt{2}\"<\/p>\n

          \u03b8.<\/strong> \"\sqrt{75} - \sqrt{12} + 5\sqrt{3}\"<\/p>\n

          \u03b9.<\/strong> \"\sqrt{24} + \sqrt{54}\"<\/p>\n

           <\/p>\n<\/div>\n<\/div>\n

          2. \u039d\u03b1 \u03c5\u03c0\u03bf\u03bb\u03bf\u03b3\u03af\u03c3\u03b5\u03c4\u03b5 \u03c4\u03b9\u03c2 \u03c0\u03b1\u03c1\u03b1\u03c3\u03c4\u03ac\u03c3\u03b5\u03b9\u03c2:<\/strong><\/em><\/span><\/p>\n

          \n
          \n

          \u03b1.<\/strong> \"\sqrt{2} (\sqrt{18} + \sqrt{8} )\"<\/p>\n

          \u03b2.<\/strong> \"\sqrt{7} (\sqrt{28} + \sqrt{63})\"<\/p>\n

          \u03b3.<\/strong> \"\sqrt{5} (\sqrt{20} + \sqrt{45})\"<\/p>\n

          \u03b4.<\/strong> \"\sqrt{3} (\sqrt{48} + \sqrt{75}) \"<\/p>\n<\/div>\n

          \n

          \u03b5. \u00a0<\/strong>\"\dfrac{\sqrt{50} + \sqrt{18} - \sqrt{200}}{\sqrt{2}}\"<\/p>\n

          \u03c3\u03c4.<\/strong> \"\dfrac{\sqrt{48} + \sqrt{27} - \sqrt{108}}{\sqrt{3}}\"<\/p>\n

           <\/p>\n<\/div>\n<\/div>\n

          \n

          \u0392\u03b9\u03b2\u03bb\u03b9\u03bf\u03b3\u03c1\u03b1\u03c6\u03af\u03b1:<\/strong> \u039c\u03b1\u03b8\u03b7\u03bc\u03b1\u03c4\u03b9\u03ba\u03ac \u0393 \u0393\u03c5\u03bc\u03bd\u03b1\u03c3\u03af\u03bf\u03c5 (\u0394\u03b7\u03bc\u03ae\u03c4\u03c1\u03b9\u03bf\u03c2 \u0391\u03c1\u03b3\u03c5\u03c1\u03ac\u03ba\u03b7\u03c2 , \u03a0\u03b1\u03bd\u03b1\u03b3\u03b9\u03ce\u03c4\u03b7\u03c2 \u0392\u03bf\u03c5\u03c1\u03b3\u03ac\u03bd\u03b1\u03c2, \u039a\u03c9\u03bd\u03c3\u03c4\u03b1\u03bd\u03c4\u03af\u03bd\u03bf\u03c2 \u039c\u03b5\u03bd\u03c4\u03ae\u03c2, \u03a3\u03c4\u03b1\u03bc\u03b1\u03c4\u03bf\u03cd\u03bb\u03b1 \u03a4\u03c3\u03b9\u03ba\u03bf\u03c0\u03bf\u03cd\u03bb\u03bf\u03c5, \u039c\u03b9\u03c7\u03b1\u03ae\u03bb \u03a7\u03c1\u03c5\u03c3\u03bf\u03b2\u03ad\u03c1\u03b3\u03b7\u03c2 \u03a5.\u03a0\u0391\u0399.\u0398.)<\/p>\n

          Creative Commons \u0391\u03bd\u03b1\u03c6\u03bf\u03c1\u03ac \u0394\u03b7\u03bc\u03b9\u03bf\u03c5\u03c1\u03b3\u03bf\u03cd – \u039c\u03b7 \u0395\u03bc\u03c0\u03bf\u03c1\u03b9\u03ba\u03ae \u03a7\u03c1\u03ae\u03c3\u03b7 – \u03a0\u03b1\u03c1\u03cc\u03bc\u03bf\u03b9\u03b1 \u0394\u03b9\u03b1\u03bd\u03bf\u03bc\u03ae 4.0 \u0394\u03b9\u03b5\u03b8\u03bd\u03ad\u03c2<\/strong><\/p>\n

          \"\"<\/p>\n","protected":false},"excerpt":{"rendered":"

          \u03a3\u03b7\u03bc\u03b5\u03b9\u03ce\u03c3\u03b5\u03b9\u03c2 \u0398\u03b5\u03c9\u03c1\u03af\u03b1\u03c2 \u03a0\u03c9\u03c2 \u03bf\u03c1\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03b7 \u03c4\u03b5\u03c4\u03c1\u03b1\u03b3\u03c9\u03bd\u03b9\u03ba\u03ae \u03c1\u03af\u03b6\u03b1 \u03b5\u03bd\u03cc\u03c2 \u03b8\u03b5\u03c4\u03b9\u03ba\u03bf\u03cd \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03cd x;; \u0397 \u03c4\u03b5\u03c4\u03c1\u03b1\u03b3\u03c9\u03bd\u03b9\u03ba\u03ae \u03c1\u03af\u03b6\u03b1 \u03b5\u03bd\u03cc\u03c2 \u03bc\u03b7 \u03b1\u03c1\u03bd\u03b7\u03c4\u03b9\u03ba\u03bf\u03cd \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03cd x \u03c3\u03c5\u03bc\u03b2\u03bf\u03bb\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03bc\u03b5 \u03ba\u03b1\u03b9 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bf \u03bc\u03b7 \u03b1\u03c1\u03bd\u03b7\u03c4\u03b9\u03ba\u03cc\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2 \u03b1 \u03c0\u03bf\u03c5 \u03cc\u03c4\u03b1\u03bd \u03c5\u03c8\u03c9\u03b8\u03b5\u03af \u03c3\u03c4\u03bf 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